The critical momentum \(p_0\)  and the power-low index \(m\)  are free parameters of our model for now.
It is important to note here that the energy in fast-mode waves has mostly a kinetic character. As the magnetic component of the fast-mode turbulence in the typical post-shock plasma is weak, it cannot amplify the magnetic field sufficiently. Therefore, in our approach other type of waves, such as streaming instabilities [1]}, [2]} or turbulent dynamos [3]} are required to provide the magnetic-field amplification at the shock, which is an important scale factor for scattering (see Sec. REF ). There is not necessarily a simple scaling between momentum and spatial diffusion [4]}. In addition, different types of turbulence may be responsible for the diffusive transport and stochastic re-acceleration [5]}. Therefore, the spatial and the momentum diffusion coefficients are independent in our model.
